Although the mechanical properties of ligament and tendon are well documented in research literature, very few unified mechanical formulations can describe a wide range of different loadings. The aim of this study was to propose a new model, which can describe tendon responses to various solicitations such as cycles of loading, unloading, and reloading or successive relaxations at different strain levels. In this work, experiments with cycles of loading and reloading at increasing strain level and sequences of relaxation were performed on white New Zealand rabbit Achilles tendons. We presented a local formulation of thermodynamic evolution outside equilibrium at a representative element volume scale to describe the tendon’s macroscopic behavior based on the notion of relaxed stress. It was shown that the model corresponds quite well to the experimental data. This work concludes with the complexity of tendons’ mechanical properties due to various microphysical mechanisms of deformation involved in loading such as the recruitment of collagen fibers, the rearrangement of the microstructure (i.e., collagens type I and III, proteoglycans, and water), and the evolution of relaxed stress linked to these mechanisms.

1.
Wang
,
J. H. C.
, 2006, “
Mechanobiology of Tendon
,”
J. Biomech.
0021-9290,
39
(
9
), pp.
1563
1582
.
2.
Woo
,
S. L.Y.
,
Abramowitch
,
S. D.
,
Kilger
,
R. C.
, and
Liang
,
R.
, 2006, “
Biomechanics of Knee Ligaments: Injury, Healing, and Repair
,”
J. Biomech.
0021-9290,
39
(
1
), pp.
1
20
.
3.
Djian
,
P.
,
Bellier
,
G.
, and
Christel
,
P.
, 1999, “
Aspects Biologiques de la Reconstruction du Ligament Croisé Antérieur par une Greffe Tendineuse Autologue
,”
Ann. Soc. fr. Arthroscopie
.
4.
De Vita
,
R.
, and
Slaughter
,
W. S.
, 2006, “
A Structural Constitutive Model for the Strain Rate-Dependent Behavior of Anterior Cruciate Ligaments
,”
Int. J. Solids Struct.
0020-7683,
43
(
6
), pp.
1561
1570
.
5.
Peña
,
E.
,
Calvo
,
B.
,
Martinez
,
M. A.
, and
Doblaré
,
M.
, 2007, “
An Anisotropic Visco-Hyperelastic Model for Ligaments at Finite Strains. Formulation and Computational Aspects
,”
Int. J. Solids Struct.
0020-7683,
44
(
3–4
), pp.
760
778
.
6.
Pioletti
,
D. P.
, 1997, “
Viscoelastic Properties of Soft Tissues: Application to Knee Ligaments and Tendons
,” Ph.D. thesis, EPFL, Lausanne, Switzerland.
7.
Pioletti
,
D. P.
, and
Rakotomanana
,
L. R.
, 2000, “
Non-Linear Viscoelastic Laws for Soft Biological Tissues
,”
Eur. J. Mech. A/Solids
0997-7538,
19
(
5
), pp.
749
759
.
8.
Fratzl
,
P.
,
Misof
,
K.
,
Zizak
,
I.
,
Rapp
,
G.
,
Amenitsch
,
H.
, and
Bernstorff
,
S.
, 1997, “
Fibrillar Structure and Mechanical Properties of Collagen
,”
J. Struct. Biol.
1047-8477,
122
(
1–2
), pp.
119
122
.
9.
Fung
,
Y. C.
, 1993,
Biomechanics: Mechanical Properties of Living Tissues
,
2nd ed.
,
Springer-Verlag
,
New York
.
10.
Magnenet
,
V.
,
Rahouadj
,
R.
,
Bacher
,
P.
, and
Cunat
,
C.
, 2008, “
Inelastic Constitutive Relations for Foamed Materials: A Statistical Approach and Its Application to Open-Cell Melamine
,”
Mech. Mater.
0167-6636,
40
(
9
), pp.
673
684
.
11.
Provenzano
,
P.
,
Lakes
,
R.
,
Keenan
,
T.
, and
Vanderby
,
R.
, Jr.
, 2001, “
Nonlinear Ligament Viscoelasticity
,”
Ann. Biomed. Eng.
0090-6964,
29
, pp.
908
914
.
12.
Ciarletta
,
P.
,
Dario
,
P.
, and
Micera
,
S.
, 2008, “
Pseudo-Hyperelastic Model of Tendon Hysteresis From Adaptative Recruitment of Collagen Type I Fibrils
,”
Biomaterials
0142-9612,
29
, pp.
764
770
.
13.
Lakes
,
R. S.
, and
Vanderby
,
R.
, Jr.
, 1999, “
Interrelation of Creep and Relaxation: A Modeling Approach for Ligaments
,”
J. Biomech. Eng.
0148-0731,
121
, pp.
612
615
.
14.
Arieby
,
R.
, 2007, “
Caractérisation Mécanique et Modélisation Thermodynamique du Comportement Anisotrope du Polyéthylène à Haute Densité. Intégration des effets d'Endommagement
,” Ph.D. thesis, Nancy-Université, Nancy, France.
15.
Mrabet
,
K.
,
Rahouadj
,
R.
, and
Cunat
,
C.
, 2005, “
An Irreversible Thermodynamic Model for Semicrystalline Polymers Submitted to Multisequence Loading at Large Strain
,”
Polym. Eng. Sci.
0032-3888,
45
(
1
), pp.
42
51
.
16.
Cunat
,
C.
, 1985, “
Approche Statistique des Propriétés Thermodynamiques des Etats Liquides et Vitreux—Relaxation des Liquides et Transition Vitreuse—Influence des Associations Chimiques
,” Ph.D. thesis, University of Nancy 1, Nancy, France.
17.
De Donder
,
T.
, 1936,
Thermodynamic Theory of Affinity: A Book of Principle
,
Oxford University Press
,
Oxford, UK.
18.
Cunat
,
C.
, 1998, “
Thermodynamic Treatment of Relaxation in Frozen-In Systems—Universality of the Fluctuation Distribution Law for Relaxation Time
,”
Z. Phys. Chem., Neue Folge
0044-3336,
157
, pp.
419
423
.
19.
Pioletti
,
D. P.
,
Rakotomanana
,
L. R.
,
Benvenuti
,
J. F.
, and
Leyvraz
,
P. F.
, 1998, “
Viscoelastic Constitutive Law in Large Deformations: Application to Human Knee Ligaments and Tendons
,”
J. Biomech.
0021-9290,
31
(
8
), pp.
753
757
.
20.
Woo
,
S. L. Y.
,
Gomez
,
M. A.
, and
Akeson
,
W. H.
, 1981, “
The Time and History-Dependent Viscoelastic Properties of the Canine Medial Collateral Ligaments
,”
J. Biomech. Eng.
0148-0731,
103
, pp.
293
298
.
21.
Johnson
,
G. A.
,
Livesay
,
G. A.
,
Woo
,
S. L. Y.
, and
Rajagopal
,
K. R.
, 1996, “
A Single Integral Finite Strain Viscoelastic Model of Ligaments and Tendons
,”
J. Biomech. Eng.
0148-0731,
118
(
2
), pp.
221
226
.
22.
Weiss
,
J. A.
, and
Gardiner
,
J. C.
, 2001, “
Computational Modeling of Ligament Mechanics
,”
Biomed. Eng. (NY)
0006-3398,
29
(
1
), pp.
1
70
.
23.
Berisio
,
R.
,
Vitagliano
,
L.
,
Mazzarella
,
L.
, and
Zagari
,
A.
, 2000, “
Crystal Structure of a Collagen-Like Polypeptide With Repeating Sequence Pro-Hyp-Gly at 1.4 A Resolution: Implications for Collagen Hydratation
,”
Biopolymers
0006-3525,
56
(
1
), pp.
8
13
.
24.
Kramer
,
R. Z.
,
Venugopal
,
M. G.
,
Bella
,
J.
,
Mayville
,
P.
,
Brodsky
,
B.
, and
Berman
,
H. M.
, 2000, “
Staggered Molecular Packing in Crystals of a Collagen-Like Peptide With a Single Charged Pair
,”
J. Mol. Biol.
0022-2836,
301
(
5
), pp.
1191
1205
.
25.
Kramer
,
R. Z.
,
Bella
,
J.
,
Brodsky
,
B.
, and
Berman
,
H. M.
, 2001, “
The Crystal and Molecular Structure of a Collagen-Like Peptide With a Biologically Relevant Sequence
,”
J. Mol. Biol.
0022-2836,
311
(
1
), pp.
131
147
.
26.
Sasaki
,
N.
, and
Odajima
,
S.
, 1996, “
Stress-Strain Curve and Young’s Modulus of a Collagen Molecule as Determined by the X-Ray Diffraction Technique
,”
J. Biomech.
0021-9290,
29
(
5
), pp.
655
658
.
27.
Vesentini
,
S.
,
Fitié
,
C. F. C.
,
Montevecchi
,
F. M.
, and
Redaelli
,
A.
, 2005, “
Molecular Assessment of the Elastic Properties of Collagen-Like Homotrimer Sequences
,”
Biomech. Model. Mechanobiol.
1617-7959,
3
(
4
), pp.
224
234
.
28.
Fessel
,
G.
,
Müller
,
R.
, and
Snedeker
,
J. G.
, 2008, “
Decorin Mediated Collagen Fibril Load Sharing in Tendon; Protein Level Models and Experiments
,”
J. Biomech.
0021-9290,
41
, p.
S314
.
29.
Redaelli
,
A.
,
Vesentini
,
S.
,
Soncini
,
M.
,
Vena
,
P.
,
Mantero
,
S.
, and
Montevecchi
,
F. M.
, 2003, “
Possible Role of Decorin Glycosaminoglycans in Fibril Force Transmission in Relative Mature Tendons: A Computational Study From Molecular to Microstructural Level
,”
J. Biomech.
0021-9290,
36
(
10
), pp.
1555
1569
.
30.
Abramowitch
,
S. D.
, and
Woo
,
S. L. Y.
, 2004, “
An Improved Method to Analyze the Stress Relaxation of Ligaments Following a Finite Ramp Time Based on the Quasi-Linear Viscoelastic Theory
,”
J. Biomech. Eng.
0148-0731,
126
(
1
), pp.
92
97
.
31.
Gentleman
,
E.
,
Livesay
,
G. A.
,
Dee
,
K. C.
, and
Nauman
,
E. A.
, 2006, “
Development of Ligament-Like Structural Organization and Properties in Cell-Seeded Collagen Scaffolds In Vitro
,”
Ann. Biomed. Eng.
0090-6964,
34
(
5
), pp.
726
736
.
32.
Cunat
,
C.
, 1996, “
Lois constitutives de matériaux complexes stables ou vieillissants, Apport de la thermodynamique de la relaxation
,”
Rev. Gen. Therm.
0035-3159,
35
(
418–419
), pp.
680
685
.
33.
Cunat
,
C.
, 2001, “
The DNLR Approach and Relaxation Phenomena. Part I: Historical Account and DNLR Formalism
,”
Mech. Time-Depend. Mater.
1385-2000,
5
(
1
), pp.
39
65
.
34.
El Alam
,
H.
,
Rahouadj
,
R.
,
Haboussi
,
M.
, and
Cunat
,
C.
, 2002, “
Extension du formalisme thermodynamique DNLR aux transformations finies
,”
Proc. Premier congrès interdisciplinaire sur les matériaux en France
, Tours, France.
35.
Onsager
,
L.
, 1931, “
Reciprocal Relation in Irreversible Process
,”
Phys. Rev.
0031-899X,
38
, pp.
2265
2279
.
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